Connected components and isometries in the metric space of all closed linear subspaces of a Hilbert space
by
J.-Ph. Labrousse
University of Nice(France)

Let H be a Hilbert space over C and let F (H) be the set of all closed linear subspaces of H. #8704;M, N #8712; F(H) set g(M, N ) = #8741;P_M #8722; P_N #8741; (known as the gap metric ) where P_M , P_N denote respectively the orthogonal projections in H on M and on N. #8704;M, N #8712; F (H) such that ker(P_M+P_N #8722; I ) = 0, #936;(M, N), the bisector of M and N , is the uniquely determined element of F(H) such that (setting #936;(M, N ) = W ):

(i) P_M P_W = P_W P_N

(ii) (P_M + P_N ) P_W = P_W(P_M+P_N ) is positive definite.

A mapping #934; of F(H) into itself is called an isometry if #8704;M, N #8712; F (H), g(M, N) = g( #934;(M), #934;(N)). In this presentation we use the notion of bisector (introduced in [1]) to determine the arcwise connected components of F(H) and the properties of isometries on that space. This leads to a number of applications to linear operators and relations.

References [1] J-Ph Labrousse, Geodesics in the space of linear relations on a Hilbert space, Proc. of the 18th OT Conference, The Theta Foundation, Bucharest, (2000) 213-234

Date received: May 31, 2008